Properties

Label 688.2.g.e
Level $688$
Weight $2$
Character orbit 688.g
Analytic conductor $5.494$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [688,2,Mod(687,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.687");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 15x^{10} + 162x^{8} + 829x^{6} + 3099x^{4} + 3654x^{2} + 3364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{8} q^{5} + \beta_{10} q^{7} + (\beta_{9} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{8} q^{5} + \beta_{10} q^{7} + (\beta_{9} + \beta_{2} + 2) q^{9} + \beta_{5} q^{11} + ( - \beta_{9} + 1) q^{13} + (2 \beta_{7} + \beta_{6} + \beta_{5}) q^{15} + (\beta_{9} + \beta_{2} - 2) q^{17} + ( - \beta_{11} - \beta_{10}) q^{19} + (2 \beta_{9} - 2 \beta_{2}) q^{21} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5}) q^{23} + (2 \beta_{9} + \beta_{2} - 2) q^{25} + ( - \beta_{11} + \beta_{3}) q^{27} + (2 \beta_{8} - \beta_{4}) q^{29} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5}) q^{31} + (\beta_{8} + \beta_1) q^{33} + ( - 6 \beta_{7} - 4 \beta_{6} + 2 \beta_{5}) q^{35} + (\beta_{8} - \beta_{4} - \beta_1) q^{37} - \beta_{10} q^{39} + (2 \beta_{9} - \beta_{2} + 2) q^{41} + (\beta_{11} - 2 \beta_{7} + \beta_{5} - \beta_{3}) q^{43} + ( - \beta_{4} + \beta_1) q^{45} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5}) q^{47} + ( - 2 \beta_{9} - 4 \beta_{2} + 7) q^{49} - \beta_{11} q^{51} + ( - \beta_{9} - 4 \beta_{2} - 1) q^{53} + ( - \beta_{10} - 2 \beta_{3}) q^{55} + ( - \beta_{9} + 6 \beta_{2} - 2) q^{57} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5}) q^{59} + (\beta_{8} - \beta_1) q^{61} + (2 \beta_{11} + \beta_{10}) q^{63} + (3 \beta_{8} - \beta_1) q^{65} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5}) q^{67} + ( - 2 \beta_{8} - \beta_{4} - \beta_1) q^{69} + (\beta_{8} + 2 \beta_{4} - \beta_1) q^{73} + ( - \beta_{11} + \beta_{10} + \beta_{3}) q^{75} + (\beta_{8} + 2 \beta_{4} + \beta_1) q^{77} + (\beta_{7} + 4 \beta_{6} - 2 \beta_{5}) q^{79} + ( - \beta_{9} + 2 \beta_{2} - 3) q^{81} + (4 \beta_{6} - \beta_{5}) q^{83} + ( - 4 \beta_{8} - \beta_{4} + \beta_1) q^{85} + (7 \beta_{7} - 2 \beta_{6} + 2 \beta_{5}) q^{87} + ( - \beta_{8} - 2 \beta_{4} + \beta_1) q^{89} + ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{3}) q^{91} + ( - \beta_{8} + 2 \beta_{4} - 2 \beta_1) q^{93} + (\beta_{7} - 5 \beta_{6} - \beta_{5}) q^{95} + ( - 2 \beta_{9} - 3 \beta_{2} + 2) q^{97} + (2 \beta_{7} + 3 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{9} + 12 q^{13} - 24 q^{17} - 24 q^{25} + 24 q^{41} + 84 q^{49} - 12 q^{53} - 24 q^{57} - 36 q^{81} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 15x^{10} + 162x^{8} + 829x^{6} + 3099x^{4} + 3654x^{2} + 3364 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1249\nu^{11} - 16326\nu^{9} - 272392\nu^{7} - 3130951\nu^{5} - 11731146\nu^{3} - 24344108\nu ) / 4323204 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -95\nu^{10} - 1026\nu^{8} - 7768\nu^{6} - 19627\nu^{4} - 23142\nu^{2} - 94232 ) / 149076 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 567\nu^{11} + 7780\nu^{9} + 84024\nu^{7} + 385479\nu^{5} + 1607348\nu^{3} + 454140\nu ) / 1441068 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 567\nu^{11} + 7780\nu^{9} + 84024\nu^{7} + 385479\nu^{5} + 1607348\nu^{3} + 3336276\nu ) / 1441068 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 726\nu^{10} + 16951\nu^{8} + 159053\nu^{6} + 905310\nu^{4} + 1772795\nu^{2} + 1559359 ) / 1080801 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -567\nu^{10} - 7780\nu^{8} - 84024\nu^{6} - 385479\nu^{4} - 1607348\nu^{2} - 1174674 ) / 720534 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3811\nu^{10} - 54410\nu^{8} - 587628\nu^{6} - 2934047\nu^{4} - 9800038\nu^{2} - 7490004 ) / 4323204 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2519\nu^{11} + 38800\nu^{9} + 419040\nu^{7} + 2398783\nu^{5} + 9457148\nu^{3} + 19539156\nu ) / 4323204 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 85\nu^{10} + 918\nu^{8} + 8258\nu^{6} + 17561\nu^{4} + 20706\nu^{2} - 226916 ) / 74538 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -5565\nu^{11} - 80807\nu^{9} - 824680\nu^{7} - 3783405\nu^{5} - 11795095\nu^{3} - 4457300\nu ) / 2161602 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3969\nu^{11} + 54460\nu^{9} + 588168\nu^{7} + 2698353\nu^{5} + 9810368\nu^{3} + 3178980\nu ) / 1441068 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + 3\beta_{7} - 4\beta_{6} - \beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{9} - 27\beta_{7} + 25\beta_{6} - 3\beta_{5} - 11\beta_{2} - 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{11} + 3\beta_{10} + 30\beta_{8} - 52\beta_{4} - 52\beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\beta_{9} + 102\beta_{2} + 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 102\beta_{11} + 45\beta_{10} - 261\beta_{8} + 397\beta_{4} - 397\beta_{3} - 45\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -409\beta_{9} + 1713\beta_{7} - 1372\beta_{6} + 486\beta_{5} - 895\beta_{2} - 1781 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -895\beta_{11} - 486\beta_{10} + 3085\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -3008\beta_{9} - 13653\beta_{7} + 10627\beta_{6} - 4629\beta_{5} - 7637\beta_{2} - 13635 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7637\beta_{11} + 4629\beta_{10} + 18282\beta_{8} - 24280\beta_{4} - 24280\beta_{3} + 4629\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1
1.42513 + 2.46840i
1.42513 2.46840i
1.18337 2.04966i
1.18337 + 2.04966i
0.564480 0.977708i
0.564480 + 0.977708i
−0.564480 0.977708i
−0.564480 + 0.977708i
−1.18337 2.04966i
−1.18337 + 2.04966i
−1.42513 + 2.46840i
−1.42513 2.46840i
0 −2.85026 0 1.84963i 0 1.55367 0 5.12398 0
687.2 0 −2.85026 0 1.84963i 0 1.55367 0 5.12398 0
687.3 0 −2.36674 0 1.91101i 0 −4.13368 0 2.60147 0
687.4 0 −2.36674 0 1.91101i 0 −4.13368 0 2.60147 0
687.5 0 −1.12896 0 3.73188i 0 4.74330 0 −1.72545 0
687.6 0 −1.12896 0 3.73188i 0 4.74330 0 −1.72545 0
687.7 0 1.12896 0 3.73188i 0 −4.74330 0 −1.72545 0
687.8 0 1.12896 0 3.73188i 0 −4.74330 0 −1.72545 0
687.9 0 2.36674 0 1.91101i 0 4.13368 0 2.60147 0
687.10 0 2.36674 0 1.91101i 0 4.13368 0 2.60147 0
687.11 0 2.85026 0 1.84963i 0 −1.55367 0 5.12398 0
687.12 0 2.85026 0 1.84963i 0 −1.55367 0 5.12398 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 687.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
43.b odd 2 1 inner
172.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.g.e 12
4.b odd 2 1 inner 688.2.g.e 12
8.b even 2 1 2752.2.g.e 12
8.d odd 2 1 2752.2.g.e 12
43.b odd 2 1 inner 688.2.g.e 12
172.d even 2 1 inner 688.2.g.e 12
344.e even 2 1 2752.2.g.e 12
344.h odd 2 1 2752.2.g.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
688.2.g.e 12 1.a even 1 1 trivial
688.2.g.e 12 4.b odd 2 1 inner
688.2.g.e 12 43.b odd 2 1 inner
688.2.g.e 12 172.d even 2 1 inner
2752.2.g.e 12 8.b even 2 1
2752.2.g.e 12 8.d odd 2 1
2752.2.g.e 12 344.e even 2 1
2752.2.g.e 12 344.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\):

\( T_{3}^{6} - 15T_{3}^{4} + 63T_{3}^{2} - 58 \) Copy content Toggle raw display
\( T_{11}^{6} + 27T_{11}^{4} + 120T_{11}^{2} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 15 T^{4} + 63 T^{2} - 58)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 21 T^{4} + 111 T^{2} + 174)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 42 T^{4} + 480 T^{2} - 928)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 27 T^{4} + 120 T^{2} + 48)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 3 T^{2} - 6 T + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{3} + 6 T^{2} - 9)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} - 81 T^{4} + 1485 T^{2} + \cdots - 3712)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 72 T^{4} + 1080 T^{2} + \cdots + 1323)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 105 T^{4} + 2271 T^{2} + \cdots + 174)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 144 T^{4} + 6480 T^{2} + \cdots + 87723)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 171 T^{4} + 5805 T^{2} + \cdots + 25056)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} - 36 T + 9)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} - 6 T^{10} + \cdots + 6321363049 \) Copy content Toggle raw display
$47$ \( (T^{6} + 141 T^{4} + 5385 T^{2} + \cdots + 49152)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 3 T^{2} - 90 T + 252)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + 180 T^{4} + 3024 T^{2} + \cdots + 6912)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 126 T^{4} + 4320 T^{2} + \cdots + 25056)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 135 T^{4} + 3564 T^{2} + \cdots + 432)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( (T^{6} + 306 T^{4} + 23544 T^{2} + \cdots + 25056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 189 T^{4} + 9801 T^{2} + \cdots + 84672)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 147 T^{4} + 5592 T^{2} + \cdots + 25392)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 306 T^{4} + 23544 T^{2} + \cdots + 25056)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 6 T^{2} - 60 T + 203)^{4} \) Copy content Toggle raw display
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